We show that relying on these linear, Gaussian techniques to identify sensors that will be used for nonlinear reconstruction can lead to costly over-sensing when the underlying manifold is low-dimensional, but the data do not lie in an equally low-dimensional subspace.
This effect is most pronounced when the most energetic (highest variance) components of the data are actually functions of less-energetic components, but not vice versa.
In such cases, the linear techniques are consistently “tricked” into sensing the most energetic components while failing to capture the important less energetic ones that can actually be used for minimal reconstruction.
These situations are not merely academic, and they actually abound in physics and in data science.
As we shall discuss in Section , the problem appears in mixing layer fluid flows and in the presence of shock waves, which are both ubiquitous in aerodynamics.
The presence of important low-energy sub-harmonic frequencies is also generic behavior after a period-doubling bifurcation, which is a common route to chaos, for instance in ecosystem collapse [1]} and cardiac arrhythmia [2]}.
In data science, the problem is most pronounced when we try to select fundamental nonlinear embedding coordinates for a data set using manifold learning techniques like kernel PCA [3]}, Laplacian eigenmaps [4]}, diffusion maps [5]}, and Isomap [6]} as we shall discuss in Section REF .
